Jacobi polynomials approach to the random search problem in one dimension

Detalhes bibliográficos
Autor(a) principal: MONROY ÁLVAREZ, Diego Alejandro
Data de Publicação: 2023
Tipo de documento: Dissertação
Idioma: eng
Título da fonte: Repositório Institucional da UFPE
Texto Completo: https://repositorio.ufpe.br/handle/123456789/51799
Resumo: Lévy processes, either flights or walks, have attracted a great deal of attention from diverse fields. They have been successfully applied to model anomalous transport phenomena in superconductors, turbulence, sunlight scattering in clouds, spectroscopy and random lasers. In ecology, there are numerous evidence that living organism often forage "non-gaussianly", a behaviour that, in theory, results in more efficient searches. Short-term deviations from normality have also been observed in financial assets prices and Lévy processes have been applied to analyse market microstructure and market friction. We address the problem of one-dimensional symmetric Lévy flights that take place in a finite interval with absorbing endpoints, i.e. the target sites. Pure Lévy flights are by no means easy to tackle analitically, hence the jump step length is sampled from a power-law (Pareto I) distribution with shape parameter 0 < α < 2 thus resembling the asymptotic heavy-tailed behaviour of the Lévy α-stable distribution. For such simplified system, closed-form expressions have been reported in the literature for the absorption probability at a specific target, the mean number of steps and the mean path length before a target is encountered, of which the last two quantities are of special interest since they are related to the mean first-passage time of Lévy flyers and walkers respectively. Those approximate closed-form expressions have been obtained by means of inversion formulae related to fractional integro-differential equations and perform reasonably well provided that the departure site is not too close to the targets and away from the Gaussian regime. This work not only intends to revisit the aforementioned approach but also to explore alternative methods, such as the spectral relationship method using classical Jacobi polynomials. This method allows the inclusion of correction terms that are difficult to handle with inversion formulae. The obtained solutions predict the simulated results more accurately and in broader ranges of the stability index and the departure site location than their inversion formulae counterparts. As a drawback, one must resort to numerical methods and regularization techniques to deal with the instability arising for the ill-conditioned nature of problem.
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spelling MONROY ÁLVAREZ, Diego Alejandrohttp://lattes.cnpq.br/7194119454388446http://lattes.cnpq.br/4321118621178584RAPOSO, Ernesto Carneiro Pessoa2023-08-07T19:29:59Z2023-08-07T19:29:59Z2023-06-13MONROY ÁLVAREZ, Diego Alejandro. Jacobi polynomials approach to the random search problem in one dimension. 2023. Dissertação (Mestrado em Física) – Universidade Federal de Pernambuco, Recife, 2023.https://repositorio.ufpe.br/handle/123456789/51799Lévy processes, either flights or walks, have attracted a great deal of attention from diverse fields. They have been successfully applied to model anomalous transport phenomena in superconductors, turbulence, sunlight scattering in clouds, spectroscopy and random lasers. In ecology, there are numerous evidence that living organism often forage "non-gaussianly", a behaviour that, in theory, results in more efficient searches. Short-term deviations from normality have also been observed in financial assets prices and Lévy processes have been applied to analyse market microstructure and market friction. We address the problem of one-dimensional symmetric Lévy flights that take place in a finite interval with absorbing endpoints, i.e. the target sites. Pure Lévy flights are by no means easy to tackle analitically, hence the jump step length is sampled from a power-law (Pareto I) distribution with shape parameter 0 < α < 2 thus resembling the asymptotic heavy-tailed behaviour of the Lévy α-stable distribution. For such simplified system, closed-form expressions have been reported in the literature for the absorption probability at a specific target, the mean number of steps and the mean path length before a target is encountered, of which the last two quantities are of special interest since they are related to the mean first-passage time of Lévy flyers and walkers respectively. Those approximate closed-form expressions have been obtained by means of inversion formulae related to fractional integro-differential equations and perform reasonably well provided that the departure site is not too close to the targets and away from the Gaussian regime. This work not only intends to revisit the aforementioned approach but also to explore alternative methods, such as the spectral relationship method using classical Jacobi polynomials. This method allows the inclusion of correction terms that are difficult to handle with inversion formulae. The obtained solutions predict the simulated results more accurately and in broader ranges of the stability index and the departure site location than their inversion formulae counterparts. As a drawback, one must resort to numerical methods and regularization techniques to deal with the instability arising for the ill-conditioned nature of problem.FACEPEOs processos de Lévy, sejam voos ou caminhadas, têm atraído muita atenção de diversos campos. Eles foram aplicados com sucesso para modelar fenômenos de transporte anômalo em supercondutores, turbulência, dispersão da luz solar em nuvens, espectroscopia e lasers aleatórios. Em ecologia, existem inúmeras evidências de que organismos vivos costumam for- ragear "não gaussianamente", um comportamento que, em teoria, resulta em buscas mais eficientes. Desvios de curto prazo da normalidade também foram observados nos preços dos ativos financeiros e os processos de Lévy foram aplicados para analisar a microestrutura e o atrito do mercado. Abordamos o problema de voos de Lévy simétricos unidimensionais que ocorrem em um intervalo finito com extremidades absorventes, ou seja, os locais de destino. Os vôos Lévy puros não são fáceis de lidar analiticamente, portanto, o comprimento do passo do salto é amostrado a partir de uma distribuição de lei de potência (Pareto I) com parâmetro de forma 0 < α < 2, assemelhando-se assim ao comportamento assintótico de cauda pesada do Lévy Distribuição α-estável. Para tal sistema simplificado, expressões de forma fechada foram relatadas na literatura para a probabilidade de absorção em um alvo específico, o número mé- dio de etapas e o comprimento médio do caminho antes de um alvo ser encontrado, dos quais as duas últimas quantidades são de interesse especial uma vez que estão relacionados com o tempo médio de primeira passagem dos voadores e caminhantes de Lévy, respectivamente. Es- sas expressões aproximadas de forma fechada foram obtidas por meio de fórmulas de inversão relacionadas a equações integrais-diferenciais fracionárias e funcionam razoavelmente bem, desde que o local de partida não esteja muito próximo dos alvos e longe do regime gaussiano. Este trabalho pretende não só revisitar a abordagem acima mencionada, mas também explorar métodos alternativos, como o método de relações espectrais usando polinômios clássicos de Jacobi. Este último permite a inclusão de termos de correção que são difíceis de lidar com fórmulas de inversão. As soluções obtidas prevêem os resultados simulados com mais precisão e em intervalos mais amplos do índice de estabilidade e da localização do local de partida do que suas contrapartes de fórmulas de inversão. Como desvantagem, deve-se recorrer a métodos numéricos e técnicas de regularização para lidar com a instabilidade decorrente da natureza mal condicionada do problema.engUniversidade Federal de PernambucoPrograma de Pos Graduacao em FisicaUFPEBrasilAttribution-NonCommercial-NoDerivs 3.0 Brazilhttp://creativecommons.org/licenses/by-nc-nd/3.0/br/info:eu-repo/semantics/openAccessFísica teórica e computacionalBuscas aleatóriasPolinômios de JacobiJacobi polynomials approach to the random search problem in one dimensioninfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesismestradoreponame:Repositório Institucional da UFPEinstname:Universidade Federal de Pernambuco (UFPE)instacron:UFPEORIGINALDISSERTAÇÃO Diego Alejandro Monroy Álvarez.pdfDISSERTAÇÃO Diego Alejandro Monroy Álvarez.pdfapplication/pdf10954725https://repositorio.ufpe.br/bitstream/123456789/51799/1/DISSERTA%c3%87%c3%83O%20Diego%20Alejandro%20Monroy%20%c3%81lvarez.pdfaafd9ce6269ea3609ade92c7395a7367MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; 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dc.title.pt_BR.fl_str_mv Jacobi polynomials approach to the random search problem in one dimension
title Jacobi polynomials approach to the random search problem in one dimension
spellingShingle Jacobi polynomials approach to the random search problem in one dimension
MONROY ÁLVAREZ, Diego Alejandro
Física teórica e computacional
Buscas aleatórias
Polinômios de Jacobi
title_short Jacobi polynomials approach to the random search problem in one dimension
title_full Jacobi polynomials approach to the random search problem in one dimension
title_fullStr Jacobi polynomials approach to the random search problem in one dimension
title_full_unstemmed Jacobi polynomials approach to the random search problem in one dimension
title_sort Jacobi polynomials approach to the random search problem in one dimension
author MONROY ÁLVAREZ, Diego Alejandro
author_facet MONROY ÁLVAREZ, Diego Alejandro
author_role author
dc.contributor.authorLattes.pt_BR.fl_str_mv http://lattes.cnpq.br/7194119454388446
dc.contributor.advisorLattes.pt_BR.fl_str_mv http://lattes.cnpq.br/4321118621178584
dc.contributor.author.fl_str_mv MONROY ÁLVAREZ, Diego Alejandro
dc.contributor.advisor1.fl_str_mv RAPOSO, Ernesto Carneiro Pessoa
contributor_str_mv RAPOSO, Ernesto Carneiro Pessoa
dc.subject.por.fl_str_mv Física teórica e computacional
Buscas aleatórias
Polinômios de Jacobi
topic Física teórica e computacional
Buscas aleatórias
Polinômios de Jacobi
description Lévy processes, either flights or walks, have attracted a great deal of attention from diverse fields. They have been successfully applied to model anomalous transport phenomena in superconductors, turbulence, sunlight scattering in clouds, spectroscopy and random lasers. In ecology, there are numerous evidence that living organism often forage "non-gaussianly", a behaviour that, in theory, results in more efficient searches. Short-term deviations from normality have also been observed in financial assets prices and Lévy processes have been applied to analyse market microstructure and market friction. We address the problem of one-dimensional symmetric Lévy flights that take place in a finite interval with absorbing endpoints, i.e. the target sites. Pure Lévy flights are by no means easy to tackle analitically, hence the jump step length is sampled from a power-law (Pareto I) distribution with shape parameter 0 < α < 2 thus resembling the asymptotic heavy-tailed behaviour of the Lévy α-stable distribution. For such simplified system, closed-form expressions have been reported in the literature for the absorption probability at a specific target, the mean number of steps and the mean path length before a target is encountered, of which the last two quantities are of special interest since they are related to the mean first-passage time of Lévy flyers and walkers respectively. Those approximate closed-form expressions have been obtained by means of inversion formulae related to fractional integro-differential equations and perform reasonably well provided that the departure site is not too close to the targets and away from the Gaussian regime. This work not only intends to revisit the aforementioned approach but also to explore alternative methods, such as the spectral relationship method using classical Jacobi polynomials. This method allows the inclusion of correction terms that are difficult to handle with inversion formulae. The obtained solutions predict the simulated results more accurately and in broader ranges of the stability index and the departure site location than their inversion formulae counterparts. As a drawback, one must resort to numerical methods and regularization techniques to deal with the instability arising for the ill-conditioned nature of problem.
publishDate 2023
dc.date.accessioned.fl_str_mv 2023-08-07T19:29:59Z
dc.date.available.fl_str_mv 2023-08-07T19:29:59Z
dc.date.issued.fl_str_mv 2023-06-13
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/masterThesis
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dc.identifier.citation.fl_str_mv MONROY ÁLVAREZ, Diego Alejandro. Jacobi polynomials approach to the random search problem in one dimension. 2023. Dissertação (Mestrado em Física) – Universidade Federal de Pernambuco, Recife, 2023.
dc.identifier.uri.fl_str_mv https://repositorio.ufpe.br/handle/123456789/51799
identifier_str_mv MONROY ÁLVAREZ, Diego Alejandro. Jacobi polynomials approach to the random search problem in one dimension. 2023. Dissertação (Mestrado em Física) – Universidade Federal de Pernambuco, Recife, 2023.
url https://repositorio.ufpe.br/handle/123456789/51799
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dc.publisher.none.fl_str_mv Universidade Federal de Pernambuco
dc.publisher.program.fl_str_mv Programa de Pos Graduacao em Fisica
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